11 CONTINUOUS MAPPINGS 45

(3.11.1) In order that f be continuous at JCG e E, a necessary and sufficient
condition is that for every neighborhood V off(x0) in E',/""*(¥') be a neigh-
borhood ofx0 in E.

(3.11.2) In order that f be continuous at x0 e E, a necessary and sufficient
condition is that, for every e > 0, there exist a d > 0 such that the relation
d(xQ, x) <S implies df(f(x0)9f(x)) < B.

These are mere restatements of the definition.

The natural injection _/F: F ->E of a subspace F of E into E (1.6.1) is
continuous. Any constant mapping is continuous.

(3.11.3) Jfx0 e E is a cluster point of a set A c E, and iffis continuous at
the point XQ , thenf(xQ) is a cluster point off (A).

For if V is a neighborhood of /(x0) *n E',/""1^') is a neighborhood of
xQ in E, hence there is y e A n/~~ *(¥'), and therefore/(y) e/(A) n V.

(3.11.4) Let f be a mapping ofE into E'. The following properties are equiva-
lent:

(a) / is continuous;

(b) for every open set A' in Ef,f~1(A) is an open set in E;

(c) for every closed set A' in E'9f~l(A') is a closed set in E;

(d) for every set A in E,/(A) c/(A).

We have seen in (3.11.3) that (a)=>(d). (d)=»(c), for if A' is closed and
A =/-1(A/), then /(A) c A' = A', hence A c/-1(A/) = A; as A c A, A is
closed. (c)=>(b) from the definition of closed sets and formula (1.5.13).
Finally (b) => (a), for if V is a neighborhood of/(x0), there is an open neigh-
borhood W cV off(xQ);f~l(^V') is an open set containing x0 and contained
in/^fV'), hence/is continuous at every point x0 by (3.11.1).

It should be observed that the direct image of an open (resp. closed) set
by a continuous mapping is not in general an open (resp. closed) set; for
instance, x -»x2 is continuous in R, but the image [0, 1 [ of the open set
]— 1, +1[ is not open; x-+ l/x is continuous in the subspace E = [1, -f oo[
of R, but the image of the closed set E is the interval ]0,1] which is not
closed in R (see however (3.17.9) and (3.20.13)). E'. A